Question 1

It is possible for an estimator of   to be inconsistent while

Accepted Answer

A) converging in probability to   . 
B) Sn  
  . 
C) unbiased. 
D) Pr   → 0. 
A) converging in probability to   . 
B) Sn  
  . 
C) unbiased. 
D) Pr   → 0. C

Question 2

E

Accepted Answer

A) is the expected value of the homoskedasticity only standard errors. 
B) =   . 
C) exists only asymptotically. 
D) =   /(n-2). 
A) is the expected value of the homoskedasticity only standard errors. 
B) =   . 
C) exists only asymptotically. 
D) =   /(n-2). B

Question 3

The large-sample distribution of   1 is

Accepted Answer

A)    (
  1-β1)
  N(0
  where νi= (Xi-μx)ui 
B)    (
  1-β1)
  N(0
  where νi= ui 
C)    (
  1-β1)
  N(0
  where νi= Xiui 
D)    (
  1-β1)
  N(0
  
A)    (
  1-β1)
  N(0
  where νi= (Xi-μx)ui 
B)    (
  1-β1)
  N(0
  where νi= ui 
C)    (
  1-β1)
  N(0
  where νi= Xiui 
D)    (
  1-β1)
  N(0
  A

Question 4

When the errors are heteroskedastic,then

Accepted Answer

A) WLS is efficient in large samples,if the functional form of the heteroskedasticity is known. 
B) OLS is biased. 
C) OLS is still efficient as long as there is no serial correlation in the error terms. 
D) weighted least squares is efficient. 
A) WLS is efficient in large samples,if the functional form of the heteroskedasticity is known. 
B) OLS is biased. 
C) OLS is still efficient as long as there is no serial correlation in the error terms. 
D) weighted least squares is efficient.

Question 5

Finite-sample distributions of the OLS estimator and t-statistics are complicated,unless

Accepted Answer

A) the regressors are all normally distributed. 
B) the regression errors are homoskedastic and normally distributed,conditional on X1,...Xn. 
C) the Gauss-Markov Theorem applies. 
D) the regressor is also endogenous. 
A) the regressors are all normally distributed. 
B) the regression errors are homoskedastic and normally distributed,conditional on X1,...Xn. 
C) the Gauss-Markov Theorem applies. 
D) the regressor is also endogenous.

Question 6

Assume that var(ui|Xi)= θ0+θ1   .One way to estimate θ0 and θ1 consistently is to regress

Accepted Answer

A)    i on
  using OLS 
B)    on
  using OLS 
C)    on
  iusing OLS 
D)    on
  using OLS but suppressing the constant ("restricted least squares") 
A)    i on
  using OLS 
B)    on
  using OLS 
C)    on
  iusing OLS 
D)    on
  using OLS but suppressing the constant ("restricted least squares")

Question 7

Asymptotic distribution theory is

Accepted Answer

A) not practically relevant,because we never have an infinite number of observations. 
B) only of theoretical interest. 
C) of interest because it tells you what the distribution approximately looks like in small samples. 
D) the distribution of statistics when the sample size is very large. 
A) not practically relevant,because we never have an infinite number of observations. 
B) only of theoretical interest. 
C) of interest because it tells you what the distribution approximately looks like in small samples. 
D) the distribution of statistics when the sample size is very large.

Question 8

Feasible WLS does not rely on the following condition:

Accepted Answer

A) the conditional variance depends on a variable which does not have to appear in the regression function. 
B) estimating the conditional variance function. 
C) the key assumptions for OLS estimation have to apply when estimating the conditional variance function. 
D) the conditional variance depends on a variable which appears in the regression function. 
A) the conditional variance depends on a variable which does not have to appear in the regression function. 
B) estimating the conditional variance function. 
C) the key assumptions for OLS estimation have to apply when estimating the conditional variance function. 
D) the conditional variance depends on a variable which appears in the regression function.

Question 9

Discuss the properties of the OLS estimator when the regression errors are homoskedastic and normally distributed.What can you say about the distribution of the OLS estimator when these features are absent?

Accepted Answer

In the initial discussion of the OLS est

Question 10

Under the five extended least squares assumptions,the homoskedasticity-only t-distribution in this chapter

Accepted Answer

A) has a Student t distribution with n-2 degrees of freedom. 
B) has a normal distribution. 
C) converges in distribution to a   distribution. 
D) has a Student t distribution with n degrees of freedom. 
A) has a Student t distribution with n-2 degrees of freedom. 
B) has a normal distribution. 
C) converges in distribution to a   distribution. 
D) has a Student t distribution with n degrees of freedom.

Question 11

If the errors are heteroskedastic,then

Accepted Answer

A) the OLS estimator is still BLUE as long as the regressors are nonrandom. 
B) the usual formula cannot be used for the OLS estimator. 
C) your model becomes overidentified. 
D) the OLS estimator is not BLUE. 
A) the OLS estimator is still BLUE as long as the regressors are nonrandom. 
B) the usual formula cannot be used for the OLS estimator. 
C) your model becomes overidentified. 
D) the OLS estimator is not BLUE.

Question 12

For this question you may assume that linear combinations of normal variates are themselves normally distributed.Let a,b,and c be non-zero constants.
(a)X and Y are independently distributed as N(a,σ2).What is the distribution of (bX+cY)?
(b)If X1,... ,Xn are distributed i.i.d.as N(a,   ),what is the distribution of     ?
(c)Draw this distribution for different values of n.What is the asymptotic distribution of this statistic?
(d)Comment on the relationship between your diagram and the concept of consistency.
(e)Let   =     .What is the distribution of   (   - a)? Does your answer depend on n?

Accepted Answer

(a)E(bX + cY)= bE(X)+ cE(Y)= a(b + c);va

Question 13

In practice,the most difficult aspect of feasible WLS estimation is

Accepted Answer

A) knowing the functional form of the conditional variance. 
B) applying the WLS rather than the OLS formula. 
C) finding an econometric package that actually calculates WLS. 
D) applying WLS when you have a log-log functional form. 
A) knowing the functional form of the conditional variance. 
B) applying the WLS rather than the OLS formula. 
C) finding an econometric package that actually calculates WLS. 
D) applying WLS when you have a log-log functional form.

Question 14

The link between the variance of   and the probability that   is within (± δ of   is provided by

Accepted Answer

A) Slutsky's theorem. 
B) the Central Limit Theorem. 
C) the Law of Large Numbers. 
D) Chebychev's inequality. 
A) Slutsky's theorem. 
B) the Central Limit Theorem. 
C) the Law of Large Numbers. 
D) Chebychev's inequality.

Question 15

Estimation by WLS

Accepted Answer

A) although harder than OLS,will always produce a smaller variance. 
B) does not mean that you should use homoskedasticity-only standard errors on the transformed equation. 
C) requires quite a bit of knowledge about the conditional variance function. 
D) makes it very hard to interpret the coefficients,since the data is now weighted and not any longer in its original form. 
A) although harder than OLS,will always produce a smaller variance. 
B) does not mean that you should use homoskedasticity-only standard errors on the transformed equation. 
C) requires quite a bit of knowledge about the conditional variance function. 
D) makes it very hard to interpret the coefficients,since the data is now weighted and not any longer in its original form.

Question 16

Slutsky's theorem combines the Law of Large Numbers

Accepted Answer

A) with continuous functions. 
B) and the normal distribution. 
C) and the Central Limit Theorem. 
D) with conditions for the unbiasedness of an estimator. 
A) with continuous functions. 
B) and the normal distribution. 
C) and the Central Limit Theorem. 
D) with conditions for the unbiasedness of an estimator.

Question 17

All of the following are good reasons for an applied econometrician to learn some econometric theory,with the exception of

Accepted Answer

A) turning your statistical software from a "black box" into a flexible toolkit from which you are able to select the right tool for a given job. 
B) understanding econometric theory lets you appreciate why these tools work and what assumptions are required for each tool to work properly. 
C) learning how to invert a 4×4 matrix by hand. 
D) helping you recognize when a tool will not work well in an application and when it is time for you to look for a different econometric approach. 
A) turning your statistical software from a "black box" into a flexible toolkit from which you are able to select the right tool for a given job. 
B) understanding econometric theory lets you appreciate why these tools work and what assumptions are required for each tool to work properly. 
C) learning how to invert a 4×4 matrix by hand. 
D) helping you recognize when a tool will not work well in an application and when it is time for you to look for a different econometric approach.

Question 18

(Requires Appendix material)This question requires you to work with Chebychev's Inequality.
(a)State Chebychev's Inequality.
(b)Chebychev's Inequality is sometimes stated in the form "The probability that a random variable is further than k standard deviations from its mean is less than 1/k2." Deduce this form.(Hint: choose δ artfully. )
(c)If X is distributed N(0,1),what is the probability that X is two standard deviations from its mean? Three? What is the Chebychev bound for these values?
(d)It is sometimes said that the Chebychev inequality is not "sharp." What does that mean?

Accepted Answer

This means that,for some distr

Question 19

The Gauss-Markov Theorem proves that

Accepted Answer

A) the OLS estimator is t distributed. 
B) the OLS estimator has the smallest mean square error. 
C) the OLS estimator is unbiased. 
D) with homoskedastic errors,the OLS estimator has the smallest variance in the class of linear and unbiased estimators,conditional on X1,…,Xn. 
A) the OLS estimator is t distributed. 
B) the OLS estimator has the smallest mean square error. 
C) the OLS estimator is unbiased. 
D) with homoskedastic errors,the OLS estimator has the smallest variance in the class of linear and unbiased estimators,conditional on X1,…,Xn.

Question 20

Consider estimating a consumption function from a large cross-section sample of households.Assume that households at lower income levels do not have as much discretion for consumption variation as households with high income levels.After all,if you live below the poverty line,then almost all of your income is spent on necessities,and there is little room to save.On the other hand,if your annual income was $1 million,you could save quite a bit if you were a frugal person,or spend it all,if you prefer.Sketch what the scatterplot between consumption and income would look like in such a situation.What functional form do you think could approximate the conditional variance var(ui   Inome)?

Accepted Answer

See the accompanying figure.va

It is possible for an estimator of to be inconsistent while

E

The large-sample distribution of 1 is

When the errors are heteroskedastic,then

Finite-sample distributions of the OLS estimator and t-statistics are complicated,unless

Assume that var(ui|Xi)= θ0+θ1 .One way to estimate θ0 and θ1 consistently is to regress

Asymptotic distribution theory is

Feasible WLS does not rely on the following condition:

Discuss the properties of the OLS estimator when the regression errors are homoskedastic and normally distributed.What can you say about the distribution of the OLS estimator when these features are absent?

Under the five extended least squares assumptions,the homoskedasticity-only t-distribution in this chapter

If the errors are heteroskedastic,then

In practice,the most difficult aspect of feasible WLS estimation is

The link between the variance of and the probability that is within (± δ of is provided by

Estimation by WLS

Slutsky's theorem combines the Law of Large Numbers

All of the following are good reasons for an applied econometrician to learn some econometric theory,with the exception of

The Gauss-Markov Theorem proves that

Economic Questions and Data

Review of Probability

Review of Statistics

Linear Regression With One Regressor

Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Linear Regression With Multiple Regressors

Hypothesis Tests and Confidence Intervals in Multiple Regression

Nonlinear Regression Functions

Assessing Studies Based on Multiple Regression

Regression With Panel Data

Regression With a Binary Dependent Variable

Instrumental Variables Regression

Experiments and Quasi-Experiments

Introduction to Time Series Regression and Forecasting

Estimation of Dynamic Causal Effects

Additional Topics in Time Series Regression

The Theory of Multiple Regression

Filters

Exam 17: The Theory of Linear Regression With One Regressor

It is possible for an estimator of to be inconsistent while

E

The large-sample distribution of 1 is

When the errors are heteroskedastic,then

Finite-sample distributions of the OLS estimator and t-statistics are complicated,unless

Assume that var(ui|Xi)= θ0+θ1 .One way to estimate θ0 and θ1 consistently is to regress

Asymptotic distribution theory is

Feasible WLS does not rely on the following condition:

Discuss the properties of the OLS estimator when the regression errors are homoskedastic and normally distributed.What can you say about the distribution of the OLS estimator when these features are absent?

Under the five extended least squares assumptions,the homoskedasticity-only t-distribution in this chapter

If the errors are heteroskedastic,then

In practice,the most difficult aspect of feasible WLS estimation is

The link between the variance of and the probability that is within (± δ of is provided by

Estimation by WLS

Slutsky's theorem combines the Law of Large Numbers

All of the following are good reasons for an applied econometrician to learn some econometric theory,with the exception of

The Gauss-Markov Theorem proves that

Economic Questions and Data

Review of Probability

Review of Statistics

Linear Regression With One Regressor

Regression With a Single Regressor: Hypothesis Tests and Confidence Intervals

Linear Regression With Multiple Regressors

Hypothesis Tests and Confidence Intervals in Multiple Regression

Nonlinear Regression Functions

Assessing Studies Based on Multiple Regression

Regression With Panel Data

Regression With a Binary Dependent Variable

Instrumental Variables Regression

Experiments and Quasi-Experiments

Introduction to Time Series Regression and Forecasting

Estimation of Dynamic Causal Effects

Additional Topics in Time Series Regression

The Theory of Multiple Regression

Filters